Inverse Problems with Poisson noise: Primal and Primal-Dual Splitting

TL;DR

This paper introduces two primal and primal-dual algorithms for solving inverse problems with Poisson noise, using a convex optimization framework that incorporates a Poisson-specific data fidelity term and a sparsity prior.

Contribution

The paper develops novel primal and primal-dual splitting algorithms tailored for Poisson noise inverse problems, with theoretical analysis and experimental validation.

Findings

Algorithms effectively handle Poisson noise in inverse problems.

Proposed methods outperform prior approaches in deconvolution tasks.

Theoretical guarantees ensure well-posedness and convergence.

Abstract

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by Poisson noise. A proper data fidelity term (log-likelihood) is introduced to reflect the Poisson statistics of the noise. On the other hand, as a prior, the images to restore are assumed to be positive and sparsely represented in a dictionary of waveforms. Piecing together the data fidelity and the prior terms, the solution to the inverse problem is cast as the minimization of a non-smooth convex functional. We establish the well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory. Experimental results on deconvolution and comparison to prior methods are also reported.